International System of Quantities

The International System of Quantities (ISQ) is a coherent system of physical quantities that serves as the foundational framework for measurements in science and technology, comprising seven base quantities—length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity—along with derived quantities formed through multiplication and division of these base quantities.^[1]^[2] Established to ensure consistency and universality in quantitative descriptions across disciplines, the ISQ underpins the International System of Units (SI), where each base quantity is associated with a corresponding base unit: the metre for length, kilogram for mass, second for time, ampere for electric current, kelvin for thermodynamic temperature, mole for amount of substance, and candela for luminous intensity.^[3] These base quantities are defined as dimensionally independent, allowing for the derivation of all other physical quantities, such as area (length squared), velocity (length per time), and force (mass times acceleration), without introducing additional independent dimensions.^[1] The ISQ promotes coherence in equations, where numerical values remain unchanged when using consistent units, facilitating international collaboration in fields like physics, engineering, and chemistry.^[2] The formal nomenclature "International System of Quantities" was introduced in the ISO 80000-1 standard published in 2009, building on earlier conventions from the SI's adoption in 1960 by the General Conference on Weights and Measures (CGPM), though the underlying quantities trace back to 19th-century metric developments.^[4]^[5] Updated in subsequent editions, including ISO 80000-1:2022, the ISQ now encompasses 13 parts of the ISO/IEC 80000 series, specifying quantities and units for specialized domains such as mechanics, thermodynamics, electromagnetism, and optics.^[2]^[6] Maintained by the International Bureau of Weights and Measures (BIPM) in coordination with the International Organization for Standardization (ISO), the ISQ ensures that advancements in fundamental constants, like the 2019 SI redefinition tying units to fixed values such as the speed of light and Planck's constant, directly enhance the precision of quantity measurements.^[1] This system is mandatory in international treaties and widely adopted globally, except in a few non-metric countries for certain applications.^[5]

Overview

Definition and Purpose

The International System of Quantities (ISQ) is the international standard system of physical quantities for use in physics, science, and technology, defined by a coherent set of base quantities and derived quantities expressed through quantity equations.^[7] It consists of seven base quantities—such as length, mass, and time—along with derived quantities formed by mathematical relations among them, providing a foundational framework independent of any specific measurement units.^[8] This system ensures that physical quantities are identified by their intrinsic properties, such as length as the extent in one dimension of a straight line, rather than by numerical values or associated units.^[7] The primary purpose of the ISQ is to establish a consistent and universal language for describing natural phenomena, enabling scientists and engineers to express physical laws and perform measurements without reliance on particular unit systems.^[8] By focusing on the relationships between quantities, the ISQ facilitates dimensional analysis, which verifies the homogeneity of equations and promotes interoperability across disciplines like mechanics, electromagnetism, and thermodynamics.^[7] This unit-independent structure underpins the formulation of fundamental scientific equations, ensuring their validity regardless of the chosen measurement scale. For instance, Newton's second law of motion is expressed in the ISQ as the quantity equation for force equaling the product of mass and acceleration:

F = m a

Here, F (force), m (mass), and a (acceleration) are all ISQ-defined quantities, with acceleration itself derived from changes in length over time intervals.^[8] This approach allows the equation to hold true conceptually, supporting precise predictions and experimental validation in diverse fields.^[7]

Relation to the International System of Units

The International System of Quantities (ISQ) serves as the foundational framework for the International System of Units (SI), providing a coherent structure of physical quantities that ensures the SI's units align directly with scientific equations without the need for scaling factors. In this system, SI units are selected such that the product or quotient of base units yields derived units that match the corresponding ISQ quantities precisely, maintaining dimensional consistency across measurements.^[9] For instance, the SI unit for velocity, the metre per second (m/s), directly corresponds to the ISQ quantity of velocity as the ratio of length to time, without any additional constants or conversion factors, illustrating the coherence inherent in the SI's design. This alignment extends to all derived quantities, where equations relating numerical values in SI units mirror the physical relationships defined in the ISQ. The seven base quantities of the ISQ form the starting point for the seven SI base units, such as the metre for length.^[9] A key distinction exists between the ISQ and SI: the ISQ defines the physical quantities themselves—what is being measured—independent of any specific measurement scale, while the SI specifies the units—how those quantities are numerically realized through standardized references tied to defining constants like the speed of light. Quantities in the ISQ, such as mass or electric current, exist as abstract properties of nature, whereas SI units provide practical, reproducible scales for quantification.^[9] In metrology, the ISQ ensures a universal language for describing physical phenomena, promoting consistency and comparability in measurements worldwide, while the SI delivers the operational standards, such as the metre defined as the distance light travels in vacuum in 1/299 792 458 of a second, enabling precise experimental realizations. This interplay supports traceability to international prototypes and constants, facilitating advancements in science and technology.^[9]

Fundamental Quantities

Base Quantities

The International System of Quantities (ISQ) is founded on seven base quantities, selected for their mutual independence and ability to serve as a foundation from which all other physical quantities can be derived through algebraic combinations, thereby encompassing the full range of observable phenomena without redundancy.^[8]^[10] These base quantities are defined in terms of their physical meanings, independent of any specific unit of measurement. The base quantities, along with their standard symbols (printed in italic type) and brief physical descriptions, are as follows:

Base Quantity	Symbol	Physical Description
Length	l or ξ	Measure of spatial extent or distance along a path.^[8]
Mass	m	Measure of the amount of matter in an object.^[8]
Time	t or τ	Duration of events or interval between occurrences.^[8]
Electric current	I	Rate of flow of electric charge.^[8]
Thermodynamic temperature	T	Measure of the average kinetic energy of particles in a system.^[8]
Amount of substance	n	Measure of the number of specified elementary entities (such as atoms or molecules).^[8]
Luminous intensity	I_v	Measure of the power emitted by a light source in a given direction, weighted by the human visual response.^[8]

In accordance with conventions established in the ISQ, symbols for quantities are single upright or italic letters (or Greek symbols), with subscripts used for specificity, while symbols for units are roman typeface; these base quantities form the irreducible set from which derived quantities are constructed.^[10]

Dimensions of Base Quantities

In the International System of Quantities (ISQ), the dimension of a physical quantity represents a fundamental aspect or property of that quantity, distinguishing it from other quantities in terms of its algebraic structure. It is conventionally denoted by enclosing the symbol for the quantity in square brackets, such as [Q] for a quantity Q, and serves as a tool for ensuring dimensional homogeneity in physical equations.^[8] The ISQ defines seven base quantities, each assigned a unique base dimension symbol, which are mutually independent and form the foundational set for expressing all other quantities. These symbols are single uppercase letters in sans-serif font, chosen for their brevity and universality in dimensional analysis. The base dimensions are as follows:

Base Quantity	Dimension Symbol
Length	[L]
Mass	[M]
Time	[T]
Electric current	[I]
Thermodynamic temperature	[Θ]
Amount of substance	[N]
Luminous intensity	[J]

For each base quantity, the dimension is simply the corresponding symbol itself; for example, the dimension of mass is [M].^[8] These base dimensions are defined to be algebraically independent, meaning no base dimension can be expressed as a product of powers of the others, which allows for a complete and non-redundant description of physical phenomena. This independence is a deliberate choice in the ISQ to facilitate coherent systems of units and simplify derivations. In practice, the base dimensions underpin dimensional analysis techniques, such as the Buckingham π theorem, which uses them to form dimensionless groups (π terms) from a set of physical variables, enabling the non-dimensionalization of equations and scaling laws in complex systems.^[8]

Derived and Special Quantities

Derived Quantities

In the International System of Quantities (ISQ), derived quantities are those that can be expressed as algebraic combinations—specifically, through multiplication and division—of the base quantities. This approach ensures that all physical quantities can be systematically related within a coherent framework, building upon the seven base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The dimensional expression for any derived quantity Q in the ISQ takes the general form [Q] = [L]^a [M]^b [T]^c [I]^d [\Theta]^e [N]^f [J]^g, where the exponents a, b, c, d, e, f, g are integers (positive, negative, or zero) corresponding to the powers of the base dimensions. This multiplicative structure reflects the fundamental way physical laws combine measurable attributes, allowing for the derivation of quantities across diverse fields of science and engineering. Prominent examples of derived quantities include velocity, defined as the ratio of length to time with dimension [L][T]^{-1}; force, as the product of mass and acceleration yielding [M][L][T]^{-2}; energy, expressed through work as force times distance resulting in [M][L]^2[T]^{-2}; and electric charge, obtained from electric current integrated over time giving [I][T]. These expressions demonstrate how derived quantities encapsulate compound relationships essential for describing motion, interactions, and electromagnetic phenomena. A key principle of coherence in the ISQ is that derived quantities maintain dimensional homogeneity in physical equations, ensuring that both sides of any valid equation possess identical dimensions. For instance, the kinetic energy equation E_k = \frac{1}{2} m v^2 balances because the dimensions of mass [M] times velocity squared ([L][T]^{-1})^2 = [L]^2[T]^{-2} yield [M][L]^2[T]^{-2}, matching the dimension of energy. This dimensional consistency underpins the reliability of scientific models and calculations across the system.

Dimensionless Quantities

Dimensionless quantities in the International System of Quantities (ISQ) are physical quantities whose dimensional expression is 1, meaning all exponents in their dimensional formula are zero, arising typically from ratios of quantities of the same dimension or from specific definitions that cancel out dimensions.^[7] These quantities are formally referred to as having dimension one or being dimensionally numbers, rather than truly "dimensionless," to emphasize their role as pure numbers within the ISQ framework.^[11] For instance, the Reynolds number, a characteristic number used in fluid mechanics, is defined as the ratio of inertial forces to viscous forces, given by

\mathrm{Re} = \frac{\rho v l}{\eta},

where \rho is density, v is velocity, l is a characteristic length, and \eta is dynamic viscosity, resulting in a dimensionless value that indicates flow regimes such as laminar or turbulent.^[12] Prominent examples include plane angle measured in radians (rad) and solid angle in steradians (sr), which are derived from ratios of arc length to radius and surface area to squared radius, respectively, yielding dimension 1.^[13] The refractive index n, defined as the ratio of the speed of light in vacuum to its phase speed in a medium at a specified frequency (n = c_0 / c), is another key example, influencing light propagation in optics without dimensional units.^[14] Similarly, the Mach number \mathrm{Ma} = v / c, comparing flow velocity v to the speed of sound c, serves as a dimensionless indicator of compressibility effects in aerodynamics.^[12] In the ISQ, dimensionless quantities are treated as a subset of derived quantities with special status, where their coherent unit is the number one (symbol 1), and named units such as the radian (for plane angle) and steradian (for solid angle), which are dimensionless derived units in the SI, to maintain consistency. This classification avoids assigning independent dimensions to angles, integrating them seamlessly into dimensional equations without altering the seven base quantities of the ISQ.^[7] These quantities play a crucial role in scientific modeling and scaling, particularly as non-dimensional groups in dimensional analysis, such as the \pi groups derived from the Buckingham \pi theorem, which enable similarity principles across different scales in phenomena like fluid flow and heat transfer.^[11] By normalizing variables, they facilitate universal predictions and reduce the complexity of physical laws, essential for engineering design and theoretical physics.^[12]

Logarithmic Quantities

Logarithmic quantities in the International System of Quantities (ISQ) are defined as dimensionless quantities expressed as the logarithm of the ratio of a physical quantity to a reference quantity of the same kind, ensuring they are inherently dimensionless by construction. This form is particularly useful for representing quantities that span wide dynamic ranges, such as intensities or amplitudes, where linear scales would be impractical.^[15] The general definition follows the form L = k \log_b (Q / Q_0), where Q is the quantity of interest, Q_0 is the reference quantity, b is the logarithmic base (typically 10 for common logarithms or e for natural logarithms), and k is a scaling constant that depends on whether the quantity relates to power or amplitude. The units for logarithmic quantities are also dimensionless, with the decibel (dB) and neper (Np) being the primary accepted units in the ISQ context. The decibel, equal to one-tenth of a bel, is defined such that for power quantities, the level is L_P = 10 \log_{10} (P / P_0) dB, and for field (amplitude) quantities like pressure or voltage, it is L_F = 20 \log_{10} (F / F_0) dB, reflecting the quadratic relationship between amplitude and power.^[15] The neper, based on the natural logarithm, is the coherent unit for such quantities when defined using \ln, with L_F = \ln (F / F_0) Np for amplitudes and L_P = \frac{1}{2} \ln (P / P_0) Np for powers; one neper corresponds to e times the reference for amplitudes. The reference quantity Q_0 must always be specified to provide physical meaning, preventing ambiguity in interpretation.^[15] Representative examples illustrate the application of logarithmic quantities across domains. In acoustics, the sound pressure level (SPL) is given by L_p = 20 \log_{10} (p / p_0) dB, where p is the root-mean-square sound pressure and the reference p_0 = 20 \, \mu\text{Pa} corresponds to the threshold of human hearing, allowing expression of sound intensities from whispers to jet engines on a compact scale. For electrical signals, the voltage level in dBV is L_V = 20 \log_{10} (V / V_0) dB, with V_0 = 1 V as the standard reference, commonly used to quantify signal strengths in audio and telecommunications equipment.^[15] In information theory, entropy—a measure of uncertainty or information content—is a logarithmic quantity, expressed in bits (using base-2 logarithm, H = -\sum p_i \log_2 p_i) or nats (using natural logarithm, H = -\sum p_i \ln p_i), where bits quantify the average information per event in binary systems. These quantities fall within the broader category of dimensionless quantities but are distinguished by their specific logarithmic form based on ratios, which provides additive properties for combining independent effects (e.g., adding dB values for cascaded systems).^[15] This logarithmic scaling emphasizes relative changes rather than absolute values, making them essential for fields requiring precise representation of ratios over orders of magnitude.

Standards and Development

Historical Development

The International System of Quantities (ISQ) traces its origins to 19th-century advancements in dimensional analysis, which provided a framework for understanding physical relations through the dimensions of quantities. Joseph Fourier introduced the core concepts of dimensional homogeneity in equations in 1822, emphasizing that physical laws must balance in terms of dimensions such as length, mass, and time.^[16] James Clerk Maxwell built on this in 1873 by formulating the mathematical theory of electromagnetic phenomena using a three-dimensional system based on length, mass, and time, thereby establishing the idea of physical quantities as products of powers of base dimensions.^[17] Lord Rayleigh advanced the method in the 1870s with his principle of dimensional similitude, enabling the derivation of functional relationships between quantities without full empirical data, influencing subsequent standardization efforts.^[18] Early 20th-century international discussions laid the groundwork for formalizing a coherent system of quantities alongside units. In 1935, the International Electrotechnical Commission (IEC) Advisory Committee on Nomenclature, in collaboration with the International Committee for Weights and Measures (CIPM), held meetings to harmonize terminology and symbols for electrical and physical quantities, addressing inconsistencies in global scientific communication.^[17] This culminated in the adoption of the International System of Units (SI) in 1960 by the 11th General Conference on Weights and Measures (CGPM), which implicitly defined an underlying system of quantities through its seven base units, promoting coherence in derived quantities.^[8] The explicit concept of a distinct system of quantities separate from units emerged in the 1990s through standards development, though the formal nomenclature "International System of Quantities (ISQ)" was introduced in 2009. The International Organization for Standardization (ISO) published ISO 1000 in 1990 (revising its 1973 edition), focusing primarily on SI units and their application, but highlighting the need for a unified treatment of quantities.^[19] This transitioned into the ISO 31 series, with ISO 31-0:1992 providing general principles for quantities, equations, and symbols, and outlining the framework of base and derived quantities independent of specific unit choices as the "system of quantities on which the SI is based".^[20] The third edition of the International Vocabulary of Metrology (VIM, 2007) incorporated this system in alignment with later standards.^[21] From 2006 to 2009, the ISO and IEC jointly developed the ISO/IEC 80000 series to supersede ISO 31 and ISO 1000, explicitly separating the definitions of quantities and their dimensions from units to enhance clarity and applicability across scientific fields, and introducing the term ISQ as shorthand for the system. The first part, ISO/IEC 80000-1:2009, outlined the general principles of the ISQ, including base quantities like length and mass, and derived quantities such as velocity and energy. Subsequent parts expanded coverage to specialized areas, ensuring the ISQ's adaptability. The 2019 redefinition of the SI by the 26th CGPM, which anchored base units to fixed values of fundamental constants like the speed of light and Planck's constant, had no direct impact on the ISQ; the quantities and their dimensional structure remained unchanged, underscoring the system's enduring stability.^[8]

Current Standards and Documentation

The ISO/IEC 80000 series, titled "Quantities and units," serves as the primary international standard defining the International System of Quantities (ISQ).^[2] Part 1 of the series, published in its second edition in December 2022, provides general information and definitions for quantities, systems of quantities, units, quantity and unit symbols, and coherent unit systems, explicitly focusing on the ISQ as the underlying framework for scientific and technological applications.^[2] The series comprises 13 parts as of 2025, covering diverse fields such as mathematics (Part 2), space and time (Part 3), mechanics (Part 4), thermodynamics (Part 5), electromagnetism (Part 6), and information science and technology (Part 13), with additional parts addressing acoustics, physical chemistry, atomic and nuclear physics, and solid-state physics.^[22] Key features of the ISO/IEC 80000 series include comprehensive compilations of quantities, their associated symbols, and dimensional expressions, ensuring uniformity across disciplines while aligning with the International System of Units (SI).^[2] These standards emphasize the ISQ's structure, which builds on seven base quantities—length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity—and derives others through multiplication and division, without introducing new base quantities.^[8] The series maintains consistency with the SI Brochure's ninth edition (2019), including its version 3.01 update from August 2024, which incorporates refinements to quantity descriptions such as angle while preserving the core ISQ-SI linkage.^[3] The standards are maintained through a joint effort by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC), primarily via ISO Technical Committee 12 (ISO/TC 12) on Quantities and units, in collaboration with IEC Technical Committee 25 for relevant parts.^[23] Recent updates include amendments such as ISO 80000-3:2019/Amd 1:2025 for space and time quantities and IEC 80000-13:2025 for information science, but no fundamental revisions to the ISQ framework have occurred as of November 2025.^[24] Accessibility to the ISO/IEC 80000 series is provided through official publications by ISO and IEC, available for purchase on their websites, with content integrated into the BIPM's SI Brochure for broader dissemination in metrology and education.^[2]^[8] For instance, Part 6 details names, symbols, and definitions for electromagnetic quantities like electric potential difference and magnetic flux density, including conversion factors where applicable, to support consistent usage in electrical engineering.